question_answer

The area bounded by the curves \[y={{\log }_{e}}x\] and \[y={{({{\log }_{e}}x)}^{2}}\] is                                               [RPET 2000]

A)            \[3-e\]                                    

B)            \[e-3\]

C)                     \[\frac{1}{2}(3-e)\]   

D)                     \[\frac{1}{2}(e-3)\]

Correct Answer: A

Solution :

           Required area \[=\int_{1}^{e}{[\log x-{{(\log x)}^{2}}]}\,dx\]            \[A=\int_{1}^{e}{\,\log x\,dx}-\int_{1}^{e}{{{(\log x)}^{2}}dx}\]               \[=[x\log x-x]\,_{1}^{e}-[x{{(\log x)}^{2}}-2x\log x+2x]\,_{1}^{e}\]               \[=[e-e-(-1)]-[e{{(1)}^{2}}-2e+2e-(2)]\]               \[=(1)-(e-2)\]\[=3-e\].